Why is this always the case A=[itex]\begin{array}{cc}1 & 6

[KingBigness]

Why is this always the case...

A=\begin{array}{cc}<br /> 1 &amp; 6 \\<br /> 4 &amp; 3 \\<br /> \end{array}

λ_{1}=7

λ_{2}=-3

A=\begin{array}{cc}<br /> 1-7 &amp; 6 \\<br /> 4 &amp; 3-7 \\<br /> \end{array}

A=\begin{array}{cc}<br /> -6 &amp; 6 \\<br /> 4 &amp; -4 \\<br /> \end{array}

Why is A_{11} and A_{12}
always a multiple of
A_{21} and A_{22}?

Is this a feature of Eigenvalues or is this done on purpose to make solving the eigenvectors easier?

 

[Fredrik]

Why is A_{11} and A_{12}
always a multiple of
A_{21} and A_{22}?

You seem to mean (A-\lambda I)_{11}, (A-\lambda I)_{12} and so on.

Ax=\lambda x implies (A-\lambda I)x=0. So if A-\lambda I is invertible, x=0. An eigenvector of A with eigenvalue \lambda is by definition a non-zero x that satisfies Ax=\lambda x. So A has an eigenvector with eigenvalue \lambda if and only if A-\lambda I is not invertible. For a matrix to be not invertible, its rows must be linearly dependent.

 

[KingBigness]

Clear as day now, thank you